This question arose from my previous question; I decided to post it as a separate question.
If we have two functors $F,G:\mathscr A\to \mathscr B$, then we say that $F(A)\cong G(A)$ naturally in $A$ if $F$ and $G$ are naturally isomorphic (Definition 1.3.12).
What's the point of such definition? According to it, $F(A)\cong G(A)$ naturally in $A$ iff $F(B)\cong G(B)$ naturally in $B$, etc. So this $A$ (or $B$) doesn't seem to be significant; because the definition is talking about $F$ and $G$ being naturally isomorphic. What's the advantage of this definition over the definition of $F$ and $G$ being naturally isomorphic?
When you have an abstract example like the one you wrote, there's not much point to it. But in practice, the notation is often messier, and you might have some construction that can be considered as a functor in several different ways, or you might have notation for the functors evaluated on some specific object but not a name for them abstractly.
For example, you might say that for any topological space $X$, the $\mathbb{Q}$-vector spaces $$H^n(X;\mathbb{Q})$$ and $$\operatorname{Hom}_{\mathbb{Q}}(H_n(X;\mathbb{Q}),\mathbb{Q})$$ are isomorphic, naturally in $X$. What does this statement mean? It means you figure out the "obvious" way in which both of these constructions can be considered as functors of $X$ (more precisely, they are the evaluation at an object $X$ of certain functors from the category of topological spaces to the category of $\mathbb{Q}$-vector spaces), and then the statement is that these two functors are naturally isomorphic. But by saying "naturally in $X$", we don't actually have to give names to these functors we are saying are naturally isomorphic. This is often very convenient for economy of writing (and of thought, so you don't have to keep track of a bunch of variables you don't really need).